We classify phases of a bosonic lattice model based on the computational complexity of classically simulating the system. We show that the system transitions from being classically simulable to classically hard to simulate as it evolves in time, extending previous results to include on-site number-conserving interactions and long-range hopping. Specifically, we construct a complexity phase diagram with easy and hard phases, and derive analytic bounds on the location of the phase boundary with respect to the evolution time and the degree of locality. We find that the location of the phase transition is intimately related to upper bounds on the spread of quantum correlations and protocols to transfer quantum information. Remarkably, although the location of the transition point is unchanged by on-site interactions, the nature of the transition point changes dramatically. Specifically, we find that there are two kinds of transitions, sharp and coarse, broadly corresponding to interacting and noninteracting bosons, respectively. Our work motivates future studies of complexity in many-body systems and its interplay with the associated physical phenomena.
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